Question

Cournot duopolists face a market demand curve given by P = 90 -Q where Q is total market demand. Each firm can produce output at a constant marginal cost of 30 per unit. There are no fixed costs. Determine the (1) equilibrium price, (2) quantity, and (3) economic profits for the total market,(4) the consumer surplus, and (5) dead weight loss.Show Work

Answer 1

P = 90 - Q = 90 - Q1 - Q2 [Since Q = Q1 + Q2]

(1) and (2)

For firm 1,

Total revenue (TR1) = P x Q1 = 90Q1 - Q1² - Q1Q2

MR1 = TR1/Q1 = 90 - 2Q1 - Q2

Equating MR1 and MC,

90 - 2Q1 - Q2 = 30

2Q1 + Q2 = 60.............(1) [Best response, firm 1]

For firm 2,

Total revenue (TR2) = P x Q2 = 90Q2 - Q1Q2 - Q2²

MR2 = TR2/Q2 = 90 - Q1 - 2Q2

Equating MR2 and MC,

90 - Q1 - 2Q2 = 30

Q1 + 2Q2 = 60.............(2) [Best response, firm 2]

Cournot equilibrium is obtained by solving (1) and (2). Multiplying (2) by 2,

2Q1 + 4Q2 = 120.........(3) and

2Q1 + Q2 = 60 ...........(1)

(3) - (1) yields:

3Q2 = 60

Q2 = 20

Q1 = 60 - 2Q2 [From (2)] = 60 - (2 x 20) = 60 - 40 = 20

Q = 20 + 20 = 40

P = 90 - 40 = 50

(3)

Profit, firm 1 = Q1 x (P - MC) = 20 x (50 - 30) = 20 x 20 = 400

Profit, firm 2 = Q2 x (P - MC) = 20 x (50 - 30) = 20 x 20 = 400

Total profit = 400 + 400 = 800

(4)

From market demand function, hen Q = 0, P = 90 (Vertical intercept of market demand curve)

Consumer surplus = Area between market demand curve & price = (1/2) x (90 - 50) x 40 = 20 x 40 = 800

(5)

Efficient outcome is obtained when P = MC.

90 - Q = 30

Q = 60

P = MC = 30

Deadweight loss = (1/2) x Difference in P x Difference in Q = (1/2) x (50 - 30) x (60 - 40) = (1/2) x 20 x 20 = 200